Computer Science Engineering (CSE) Exam > Computer Science Engineering (CSE) Questions > Which of the following statements are true?S1...
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Which of the following statements are true?
S1: Inserting n elements into empty binary heap takes O(n) time.
S2: The most efficient algorithm take O(m+n) time for finding number of connected components in undirected graph of n vertex and m edges.
- a)Only S1
- b)Only S2
- c)Both S1 and S2
- d)Neither S1 nor S2
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
Which of the following statements are true?S1: Inserting n elements in...
Understanding S1: Inserting n elements into an empty binary heap
Inserting n elements into an empty binary heap can indeed be done in O(n) time. Here’s how:
- Heap Property: A binary heap maintains a complete binary tree structure where each parent node is either greater than (max-heap) or less than (min-heap) its children.
- Insertion Process: Each insertion involves adding an element at the end and then “bubbling up” to maintain the heap property. This would typically take O(log n) time for each insertion.
- Optimal Approach: However, if all n elements are inserted simultaneously using a method called "heapify" or "build heap," the process is performed in O(n) time. This is because the majority of the elements are already in the correct position, and adjustments require less time.
Therefore, S1 is true.
Understanding S2: Finding connected components in an undirected graph
The statement regarding the time complexity for finding the number of connected components in an undirected graph is also accurate:
- Graph Representation: An undirected graph can be represented using adjacency lists, which allows efficient traversal.
- Traversal Techniques: The most efficient algorithms for this task are Depth-First Search (DFS) or Breadth-First Search (BFS). Each traversal explores all vertices and edges.
- Time Complexity: The time complexity for traversing all vertices (n) and edges (m) in an undirected graph is O(m + n). This covers marking nodes and counting components effectively.
Thus, S2 is true.
Conclusion
Both statements S1 and S2 are correct. Therefore, the correct answer is option 'C': Both S1 and S2 are true.
Inserting n elements into an empty binary heap can indeed be done in O(n) time. Here’s how:
- Heap Property: A binary heap maintains a complete binary tree structure where each parent node is either greater than (max-heap) or less than (min-heap) its children.
- Insertion Process: Each insertion involves adding an element at the end and then “bubbling up” to maintain the heap property. This would typically take O(log n) time for each insertion.
- Optimal Approach: However, if all n elements are inserted simultaneously using a method called "heapify" or "build heap," the process is performed in O(n) time. This is because the majority of the elements are already in the correct position, and adjustments require less time.
Therefore, S1 is true.
Understanding S2: Finding connected components in an undirected graph
The statement regarding the time complexity for finding the number of connected components in an undirected graph is also accurate:
- Graph Representation: An undirected graph can be represented using adjacency lists, which allows efficient traversal.
- Traversal Techniques: The most efficient algorithms for this task are Depth-First Search (DFS) or Breadth-First Search (BFS). Each traversal explores all vertices and edges.
- Time Complexity: The time complexity for traversing all vertices (n) and edges (m) in an undirected graph is O(m + n). This covers marking nodes and counting components effectively.
Thus, S2 is true.
Conclusion
Both statements S1 and S2 are correct. Therefore, the correct answer is option 'C': Both S1 and S2 are true.
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Community Answer
Which of the following statements are true?S1: Inserting n elements in...
Given,
S1: True
Inserting n elements in an empty binary heap takes O(n) time because the build heap method takes O(n).
S2: True
BFS algorithm takes O(m + n) time to find the number of connected components in undirected graphs of n vertices and m edges.
Hence, the correct option is (C).
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